With exploration and development of shale oil and gas, the wellbore instability problem in shale interval has drawn more and more attention^[1,2]. The anti-collapse drilling fluid technology taking advantage of the “synergistic effect of multiple mechanisms” has been widely used in the drilling of shale formation^[3]. That involves improving the stability of shale borehole by “sealing, inhibiting and chemical activity balance”. But how to accurately calculate the safe drilling mud density window considering shale hydration stress and related strength damage when interacting with drilling fluid is still a key problem waiting to be solved urgently. Compared with the method to characterize the macro hydration deformation by statistically analyzing from the microscopic point of view and then upscale^[4,5,6], shale macroscopic constitutive model analysis is much simpler^[7,8,9]. Heidug, Ghassemi and Zhou et al. proposed the chemo-poro-elastic constitutive equation of shale and concluded that the effective stress of shale was influenced by shale deformation, pore pressure and variation of pore fluid chemical potential^[7,8,9]. Zhang et al.^[10,11] observed the hydration induced fracture propagation with CT scanning and found that the variation of hydration stress was not consistent with what Heidug-Wong generalized Biot effective stress equation described^[7]. In addition, the hydration of clay minerals can cause the propagation of fractures and strength damage of shale^{[10,11,12,13,14]}, which hasn’t been considered by the chemo-poro- mechanical model.

In this work, based on the variation law of hydration stress, the Heidug-Wong generalized Biot effective stress equation^[7] has been corrected, and the Weibull statistical model is used to describe the hydration strain-related strength damage. Considering the drilling fluid sealing layer on the shale, by solving pore fluid flowing and solute diffusion equations, the pore pressure and solute concentration at the borehole wall can be calculated, and then shale safe drilling mud density window can be predicted. The method can be used to accurately calculate the collapse pressure, fracturing pressure and its variation with time in the shale section, to provide basis for the design of anti-collapse drilling fluid.

1.1.1. Width of hydration induced fracture in meso scale and variation pattern of hydration stress

As shown in Fig. 1, Griffith fracture was used to model hydration induced fracture under the effect of hydration swelling stress and confining stress^[11].

Hydration swelling stress (\({{\sigma }_{\text{hf}}}\)) acts as tensile stress in fracture, which can overcome fracture closing pressure (\({{\sigma }_{\text{c}}}-\alpha p\)) and induce fracture propagation. According to Griffith principle, hydration induced fracture width can be expressed as:

Ignoring the influence of pore pressure, then

Zhang et al.^[10-11] observed propagation of hydration induced fractures in shale under the conditions of 0.03 MPa pore pressure, 0.10 MPa confining pressure and 0.03 MPa pore pressure, 20.0 MPa confining pressure, respectively. Two main trends of fracture propagation were observed, (1) \(\Delta y\left( x \right)\) kept decreasing, then \({{\sigma }_{\text{hf}}}\) was smaller than \({{\sigma }_{\text{c}}}\) and \({{\sigma }_{\text{hf}}}\) kept decreasing; (2) \(\Delta y\left( x \right)\) firstly increased and then decreased, then firstly \({{\sigma }_{\text{hf}}}\) was larger than \({{\sigma }_{\text{c}}}\), and with time going, \({{\sigma }_{\text{hf}}}\) turned smaller than \(σ_c\) and \({{\sigma }_{\text{hf}}}\) kept decreasing. In both processes, \(σ_c\) was constant and it could be inferred that \({{\sigma }_{\text{h}}}\) was highest at the initial stage and then decreased with time. As Heidug et al.^[7] reported, the main driving force for shale hydration swelling was chemical potential variation in clay mineral spacing. At first, chemical potential difference between pore fluid and clay mineral spacing was the highest, resulting in highest hydration stress. With transport of solute and diffusion of water molecules, the chemical potential difference between pore fluid and clay mineral spacing decreases, and accordingly the variation of chemical potential in clay mineral spacing decreases, leading to reduction of hydration stress. Then, it can be inferred,

1.1.2. Solute transport in shale and residual hydration swelling pressure

Shale contains a lot of clay minerals (Fig. 2a). Water and solute molecules can transport through multi-scale pores in shale (Fig. 2b), including spacing between clay mineral layers in nano-scale and macro-pores^[15]. Osmotic flow of fluid in macro-pores is influenced by shale membrane efficiency, and when the membrane efficiency is equal to 0, there is no osmotic flow. But for clay mineral spacing, transport of solute and water molecules driven by chemical potential difference will result in hydration swelling stress and induce increase of clay mineral spacing. With chemical potential difference decreasing, hydration swelling stress will decrease until it is balanced by confining stress (Fig. 2c). At this point, transport of solute and water molecules are stopped and chemical difference between pore fluid and clay mineral spacing keep constant, then the residual hydration swelling stress in clay minerals spacing is the residual swelling pressure^[15,16].

Fick's law can be used to characterize the diffusion of solute in shale^[17,18]:

Considering the obstruction of external stress to transport of water and solute molecules in the clay mineral spacing, ${{D}_{\text{e}}}$can be expressed as^[17,18,19]:

1.1.3. Correction of generalized Biot effective stress equation for shale

The generalized Biot effective stress equation proposed by Heidug-Wong is^[7]:

Integrating items by t at both sides of equation 6:

Assuming only one kind of solute in the pore fluid can cause hydration swelling of shale. According to equation 3, considering the residual hydration swelling pressure, the following equation can be used to describe the hydration stress:

As Zhang et al.^[10-11] observed, only a few hydration- induced fractures kept open in the shale. Therefore, the effect of residual hydration swelling pressure on effective stress can be corrected with the factor \[{{\alpha }_{\text{d}}}\]. Then, equation 7 can be modified as:

Hydration swelling can induce fracture propagation in shale and thus strength damage^{[10,11,12,13,14]}. Different theoretical models were proposed to describe the relationship between meso scale fracture generation and macroscopic strength damage of rock^{[20,21,22,23,24,25]}, among which Weibull statistical model has been widely used^[23,24,25]. The experimental results of Zhang et al.^[11] show with shale hydration swelling strain increasing from 7.16×10^-4 to 1.662×10^-3 and 3.170×10^-3, the Mancos shale strength, which was 65.7 MPa under 3.5 MPa confining pressure before hydration, reduced from 46.6 MPa to 31.4 MPa and then 0 MPa (broken), respectively. Hence, hydration swelling strain can be considered as element strength parameter, then Weibull statistical damage model of shale related to hydration strain can be expressed as:

Under interaction with drilling fluid, the shale failure under in-situ stress can be regarded as two-stage loading by hydration swelling stress and shear stress, and the total shale damage can be expressed as ^[25]:

Assuming shale strength damage was 0 before hydration, then

According to Heidug-Wong^[7], there was local physical-chemical equilibrium during shale hydration process, and the hydration rate of shale was much faster than strain rate. Then, it can be assumed that the pore pressure and external constraints are constant during the hydration swelling process, and the hydration swelling volumetric strain can be expressed as:

Assuming only one kind of solute in the pore fluid can cause shale hydration swelling, if\[{{\omega }_{1}}\text{=}\frac{\alpha -1}{{{G}_{\text{K}}}}{{\omega }_{\text{k0}}}\frac{\partial \mu }{\partial c}\], inte-grate both sides of equation 13 by t, and include equation 10, then:

It can be seen from equation 14 that with the amount of solute or water molecules increasing in shale, the strength damage increases.

Asphalt, high molecular polymer can adhere to the surface of the shale. Besides, the polyalcohol, silicate, aluminum-based stabilizer, etc. can form precipitates plugging pores in shale at the front edge of the filtrate. Therefore, the drilling fluid sealing layer is composed of adhesion layer and surface shale filled with plugging agents, which is like none-perfect semipermeable membrane, and is much denser than the original shale. Based on these facts, the model of the wellbore is established, as shown in Fig. 3 and there is a sealing layer on the borehole wall, which can affect the transport of water and solute from the drilling fluid to the borehole wall, thereby affecting the pore pressure and solute concentration at the borehole wall.

Mass balance equation for pore fluid flowing in shale is^[2]:

Based on the continuity equation, ignoring the influence of chemical potential, solid deformation, and solute adsorption on fluid flow, the pore fluid mass balance equation of the sealing layer can be expressed as:

Assuming that the relationship between permeability and porosity of the sealing layer is consistent with that of shale, then the porosity of sealing layer can be expressed as^[14]:

As shown in Fig. 3, the boundary conditions for pore fluid flowing are:

Mass balance equation for solute diffusion in shale is^[2]:

Mass balance equation of solute in sealing layer is:

According to equations 4 and 5, assuming that the diffusion correction coefficients and pore geometric characteristic parameter of the sealing layer are consistent with those of shale, then the diffusion coefficients of the sealing layer can be expressed as follows:

At infinite distance, as the content of K⁺ or other prohibition agents in shale pore fluid are very low and negligible, the diffusion boundary conditions are:

According to strain equivalence hypojournal proposed by Lemaitre J^[26], considering strength damage, the modified effective strength equation can be expressed as:

\[\sigma _{i\text{,}j}^{\text{*}}=\frac{{{\sigma }_{i\text{,}j}}-\left[ \alpha p-\left( {{\omega }_{\text{0}}}\frac{\partial c}{\partial t}+{{\alpha }_{\text{d}}}{{\pi }_{\text{d}}} \right) \right]{{\delta }_{i\text{,}j}}}{1-D}\]

Mohr-Coulomb failure criterion is one of the shear failure strength criteria widely used in borehole stability analysis^[23], which can be expressed as

Tensile failure strength criterion can be expressed as:

Equations (15) and (16) and equations (19) and (20) are solved by Comsol finite element software to obtain borehole wall pore pressure and solute concentration, respectively. Thereafter, strength damage related to solute concentration is calculated by equation (14). Based on the in-situ stress, the stress distribution around the well and the principal stress of the borehole wall are calculated by using MATLAB software. The collapse pressure and fracturing pressure are calculated based on the shear and tensile failure criteria considering the hydration swelling stress and strength damage. The flow chart is shown in Fig. 4.

The anti-collapse drilling fluid mainly improves wellbore stability by making use of its sealing and inhibition properties. If there is osmotic flow in shale, the concentration of inorganic salt in the drilling fluid can be increased to induce pore fluid to flow out of shale into drilling fluid, to reduce pore pressure and improve wellbore stability. According to the test results of plugging, diffusion and swelling parameters (Table 1) of shale in literatures [2, 15-18, 23-25, 27-29], the method presented in this paper was used to calculate the safe drilling mud density window. The influence of drilling fluid anti-collapse performance on borehole wall pore pressure, solute concentration, hydration swelling stress, hydration strength damage and safe drilling mud density window were analyzed.

3.1.1. Influence drilling fluid sealing performance on transmission of pore pressure at borehole wallIf there is no osmotic flow in shale, the parameters which can influence borehole wall pore pressure are the thickness and permeability of the sealing layer. The parameters in Table 1 were used to calculate the pore pressure at the borehole wall at different sealing layer permeabilities and thicknesses (Fig. 5). (1) The pore pressure is positively correlated with time, it rises rapidly in the initial stage, after inflection points appearing on the curve within 1.5 days, then it rises slowly. (2) When the thickness of the sealing layer is constant, the lower the permeability of the sealing layer, the lower the pore pressure is. (3) The effect of sealing layer thickness on pore pressure change decreases significantly with the increase of sealing layer permeability. For example, when K_s=1×10^-10 μm², the change of sealing layer thickness has little influence on borehole pore pressure, and after 10 days the three curves of pore pressure at different sealing layer thicknesses nearly coincide. (4) When the permeability of the sealing layer is constant, the smaller the thickness of the sealing layer, the higher the pore pressure will be. It can be seen that permeability and thickness of sealing layer are the key factors controlling pore pressure transmission.

3.1.2. Influence of osmotic flow on borehole wall pore pressureAccording to the pore fluid flow and solute transport model, if the activity of drilling fluid (inorganic salt concentration) remains constant, the factor influencing borehole pore pressure include thickness, permeability and membrane efficiency of sealing layer. We set p₀ to be 30 MPa, d_s to be 1, 2 mm, sealing layer membrane efficiency to be 0.3, 0.5. Other parameters were the same as listed in Table 1, the borehole wall pore pressure was calculated (Table 2 and Fig. 6).

As shown in Fig. 6, if there is osmotic flow in shale, under the influence of the sealing layer: (1) The pore pressure of borehole wall gradually decreases to the minimum with time, then increases gradually and tends to be stable. (2) When the thickness of the sealing layer is constant, the lower the permeability of the sealing layer, the larger the reduction magnitude of borehole pore pressure, and the slower the increase rate of the pore pressure after dropping to the minimum will be. For example, when d_s=1 mm, σ_ms=0.3, if the sealing layer permeability reduces from 1.00×10^-10 μm² to 0.10×10^-10 μm², and 0.01×10^-10 μm², the minimum pore pressure will reduce from 20.39 MPa to 16.86 MPa and 14.44 MPa, respectively. Accordingly, after 10 days, the borehole wall pore pressure will reduce from 21.87 MPa to 20.76 MPa and 15.36 MPa. (3) When the thickness of sealing layer increases, both the rates of pore pressure decreasing and then increasing will slow down, and the lowest value of pore pressure slightly decreases. For example, when K_s=0.10×10^-10 μm², d_s increases from 1 mm (p5) to 2 mm (p7), the lowest value of pore pressure decreases from 16.86 MPa to 16.84 MPa, and 10 days later pore pressure decreases from 20.76 MPa to 19.67 MPa. (4) The effect of increasing sealing layer membrane efficiency on borehole wall pore pressure decreases with the drop of sealing layer permeability. For example, when K_s=1.00×10^-10 μm², σ_ms increases from 0.3 to 0.5, the minimum pore pressure decreases by Δp_D1=0.03 MPa when d_s=1 mm, and decreases by Δp_D2=0.16 MPa when d_s=2 mm. When K_s=0.10×10^-10 μm² and 0.01×10^-10 μm², the change of membrane efficiency has little influence on borehole pore pressure, and the curves of pore pressure almost coincide. It can be seen that the permeability and thickness of the sealing layer are the key factors affecting the pore pressure.

According to the generalized effective stress equation, the hydration swelling stress of shale at borehole wall \[{{\sigma }_{\text{he}}}\] can be expressed as:

According to the solute diffusion equation, generalized effective stress equation, and shale hydration strength damage equation, if the pore pressure remains constant, the factors influencing shale hydration stress at borehole wall include solute concentration, shale hydration swelling coefficient and residual swelling pressure; and the factors influencing shale hydration strength damage (equation 14) include γ and m. If the value of m is 1.5 and γ ranges from 0.005 to 0.030, the D will range from 0 to 0.9945.

Given the diffusion coefficient of solute in shale of 1× 10^-10 m²/s, and the diffusion coefficient in the sealing layer of (0.01-1.00)×10^-11 m²/s, the thickness of the sealing layer of 1-3 mm, the concentration of inhibitor in shale of zero, and the concentration of inhibitor (KCl) in drilling fluid of 100 g/L, the solute concentrations at the borehole wall with different diffusion coefficients were calculated (Fig. 7). The results show that: (1) The solute concentration is positively correlated with time, rise rapidly at the initial stage, and then rises slowly after inflection point occurring about 1 day later. (2) The lower the sealing layer diffusion coefficient, the lower the solute concentration at the borehole wall is, and with time going, solute concentration varies greatly at the initial stage with diffusion coefficient of sealing layer and tends to be stable in the later stage. (3) The thicker the sealing layer, the lower the solute concentration at the borehole wall will be, similarly, with the time going, the thickness of sealing layer has more significant effect on solute concentration at the borehole wall in the initial stage, and later the effect tends stable. Clearly, decrease of sealing layer diffusion coefficient can reduce the solute diffusion. According to equation (5), the effective diffusion coefficient of solute is related to external constrains, with the increase of external compressive stress, solute diffusion reduces.

Fig. 8 shows the influence of diffusion coefficient on hydration swelling stress in shale (at hydration swelling coefficient ω₀ of 2.0). Initially, no solute or water diffusion occurs. Although the diffusion coefficients are different, the hydration swelling stress is the same as the shale property and the solute concentrations on both sides are the same, thus the chemical potential differences on both sides are the same. Then, as time goes on, the hydration swelling stress decreases gradually and tends to be constant. 10 days later, the difference between hydration swelling stresses at D_es=1×10^-12 m²/s and D_es= 1×10^-13 m²/s is 0.31 MPa, while the difference between hydration swelling stresses at D_es=1×10^-11 m²/s and D_es=1×10^-12 m²/s is 2.58 MPa, which is about 8 times the former.

Fig. 9 shows the calculated results of shale strength damage at different diffusion coefficients and damage coefficients γ with m=1.5. It can be seen: (1) Rock strength damage is positively correlated with time, rises rapidly in the initial stage, and then slowly after inflection point occurring in about 1 day. (2) The greater the damage coefficient γ, the greater the strength damage is; and the damage coefficient will have stronger impact on the shale strength damage as time goes on. (3) When the diffusion coefficient decreases, as the solute transport and water diffusion turn slower, the hydration swelling strain decreases, and thus the resulted shale strength damage reduces.

According to the discussion above, the sealing and inducing osmotic flow performance of drilling fluid mainly influence the pore pressure transmission and solute diffusion at the borehole wall, which can be characterized by the permeability, diffusion coefficient and thickness of sealing layer. The thickness of sealing layer can be adjusted in a small range, and the thinner sealing layer is more tight and lower in permeability^[2]. According to equation (17) and equation (21), permeability of the sealing layer is the major parameter that influence the pore pressure and solute concentration at the borehole wall. The inhibition performance of drilling fluid mainly affects the effective stress and shale strength damage at the borehole wall, which can be characterized by shale hydration swelling coefficient, residual hydration swelling pressure and damage coefficient. Therefore, the permeability, shale hydration swelling coefficient, strength damage coefficient and residual hydration swelling pressure of drilling fluid sealing layer are the main parameters influencing the safe drilling mud density window of shale borehole wall.

With the parameters in Table 1, setting d_s=1 mm, σ_ms= 1.5σ_m, m=1.5, and assuming that with the inhibition performance of drilling fluid improving, ω₀, γ and α_dπ_d decrease at the same time and at the same magnitude. The parameters combinations as shown in Table 3 were designed to calculate the safe drilling mud density window at different sealing performances (K_s), inhibition performances (ω₀, γ, α_dπ_d) and inducing osmosis flow performances (σ_ms). The results are shown in Fig. 10.

It can be known that: (1) The collapse pressure is positively correlated with the time, rises rapidly in the initial stage, and then rises slowly after the inflection point appears about 1 day later; the fracturing pressure is negatively correlated with time, decreases rapidly at first, and then decreases slowly after the inflection point occurs about 1 day later. (2) If the inhibition performance of drilling fluid is improved, ω₀, γ and α_dπ_d are reduced, the collapse pressure will reduce greatly, and the fracturing pressure will slightly increase. For example, when K_s=1.0×10^-10 μm², if there is no osmosis flow, with parameter combination C3 turning into parameter combinations C2 and C1, after 10 days, the safe drilling mud density window varies from no window (collapse pressure of 2.07 g/cm³, and the fracturing pressure of 1.98 g/cm³) to 1.56-2.01 g/cm³, and 1.10-2.03 g/cm³. (3) When the permeability of the sealing layer decreases, the collapse pressure decreases, and the fracturing pressure increases, widening the safe drilling mud density window. With the increase of collapse pressure equivalent density, the effect of sealing layer permeability on collapse pressure increases, and more reduction of the collapse pressure equivalent density will be induced with sealing layer permeability decreasing. For example, when the permeability of the sealing layer reduces from 1.0×10^-10 μm² to 1.0×10^-11 μm², if without osmotic flow, the safe drilling mud density window of 1.10-2.03 g/cm³ for the parameter combination C1 after 10 days would widen to 1.10-2.08 g/cm³ for parameter combination C7, and the safe drilling mud density window of parameter combination C3 would widen from no (collapse pressure being 2.07 g/cm³, fracturing pressure being 1.98 g/cm³) to 1.98 to 2.03 g/cm³. (4) If there is osmotic flow in shale, osmotic flow of fluid from shale to drilling fluid can reduce collapse pressure, and increase fracturing pressure, widening the window of safe drilling mud density. The lower the permeability of sealing layer, the greater its influence will be. For example, when sealing layer permeability is 1.0×10^-10 μm², under the influence of osmotic flow from shale to drilling fluid, the safe drilling mud density window (1.10-2.03g/cm³) of parameter combination C1 would widen to 1.09-2.04 g/cm³ of parameter combination C4 10 days later; when the sealing layer permeability is 0.1×10^-10 μm², under the influence of osmotic flow from shale to drilling fluid, the safe drilling mud density window (1.10-2.08 g/cm³) of parameter combination C7 would widen to 1.02-2.15 g/cm³ of parameter combination C10 after 10 days.

It can be seen that, for shale with significant strength damage due to hydration swelling, the improvement of drilling fluid inhibition can significantly reduce equivalent density of borehole wall collapse pressure. The improvement of drilling fluid sealing performance can reduce the permeability of sealing layer, reduce pore pressure transmission and solute diffusion, and thus prolong the collapse period and widen the window of safe drilling mud density. Moreover, the effecting magnitude of sealing layer permeability on collapse pressure equivalent density increases with the increase of collapse pressure equivalent density. Osmotic flow from shale into drilling fluid can reduce pore pressure and increase effective stress, thus making the collapse pressure equivalent density drop and the fracturing pressure equivalent density increase, and thus the window of safe drilling mud density widen. Moreover, the lower the sealing layer permeability, the greater the influence of osmotic flow on collapse pressure will be.

ω₀, γ and α_dπ_d are three parameters related to drilling fluid inhibition performance. When γ is kept unchanged ω₀ is 0, 1, 2, α_dπ_d is 0, 0.5, 1.0 MPa, K_s is 1.0×10^-11 μm², and there is no osmotic flow, the safe drilling mud density window is calculated, as shown in Fig. 11. It can be seen from the figure that: (1) The increase of ω₀ causes the increase of collapse pressure and the decrease of fracturing pressure. With time going, the influence of ω₀ variation on collapse pressure and fracturing pressure decreases gradually, and the influence can be ignored after 0.25 days. For example, with the increase of ω₀, the initial safe drilling fluid density window for parameter combination C13 changes from 1.09-2.9 g/cm³ to 1.19-2.71 g/cm³ for parameter combination C14 and 1.28-2.53 g/cm³ for parameter combination C9, and the safe drilling mud density windows for parameter combination C13, C14 and C9 are the same 1.98-2.03 g/cm³ after 0.25 days. (2) The increase of α_dπ_d results in the increase of collapse pressure and the decrease of fracturing pressure. For example, the safe drilling mud density window for parameter combination C15 of 1.94-2.05 g/cm³ changes into 1.96-2.04 g/cm³ for parameter combination C16 and 1.98-2.03 g/cm³ for parameter combination C9 10 days later. Compared with the effecting magnitude of drilling fluid inhibition performance (ω₀, γ, α_dπ_d) on collapse pressure equivalent density shown in Fig. 10, the change of ω₀ and α_dπ_d has smaller influence on the safe drilling mud density window. That is to say the decrease of shale strength damage (γ ) due to the improvement of drilling fluid inhibition performance is the decisive factor affecting collapse pressure equivalent density.

Improving drilling fluid sealing performance can reduce pore pressure transmission and solute diffusion at the borehole wall. The inhibition performance of drilling fluid, especially the inhibition to shale strength damage, is crucial to borehole wall collapse pressure for shale interval with significant hydration swelling property. Improving sealing and inhibition performance of drilling fluid can reduce collapse pressure and enhance fracturing pressure, and thus widen the safe drilling mud density window and prolong wellbore collapse time. The higher the collapse pressure, the higher the influence of drilling fluid sealing performance on it will be.

Osmotic flow of fluid from shale to drilling fluid can widen the safe drilling mud density window, and the better the drilling fluid sealing performance, the greater the influence of the osmotic flow on the safe drilling mud density window will be.

By modifying the generalized Biot effective stress equation and using Weibull statistical model to describe shale strength damage related to hydration strain, the method to calculate safe drilling mud density window has been established. The method can analyze the relationship between collapse pressure, fracturing pressure and anti-collapse performance of drilling fluid, and therefore can be used for accurate calculation of safe drilling mud density window and optimization of drilling fluid performance.

Nomenclature

a—constant, -24.6;

c—molar concentration of solute, mol/m³;

c₀—initial molar concentration of solute in shale pore fluid, mol/m³;

c_f—molar concentration of solute in drilling fluid, mol/m³;

c_id—solute content in clay mineral spacing, mol/m³;

c_ip—molar concentration of solute in sink pore fluid, mol/m³;

c_is—molar concentration of solute in source pore fluid, mol/m³;

c_k—molar concentration of k component solute, mol/m³;

c_s—molar concentration of solute in sealing layer pore fluid, mol/m³;

C—shale cohesion force, MPa;

C_ac—average concentration of solute ions, mol/m³;

C_sD—average concentration of solute, mol/m³;

d—clay mineral spacing, 0.1 nm;

d_s—sealing layer thickness, mm;

D—hydration swelling damage, dimensionless;

D_a—free diffusion coefficient of solute in pores, m²/s;

D_e—effective diffusion coefficient, m²/s;

D_es—effective diffusion coefficient of solute in pores of sealing layer, m²/s;

D_m—total damage caused by hydration swelling and mechanical loading, dimensionless;

D_ms—mechanical loading damage, dimensionless;

D_s—free diffusion coefficient of solute in water, m²/s;

g—gravitational acceleration, 9.8 m/s²;

G_K—volume modulus of shale, MPa;

G_Kf—volume modulus of fluid, MPa;

G_Km—volume modulus of shale matrix, MPa;

G_s—shear modulus of shale, MPa;

J_da—solute diffusion flux, mol/(m²•s);

i, j—component number;

k—solute component number;

K—permeability, 10³ μm²;

K_h—flowing efficiency, m/s;

K_s—permeability of sealing layer, 10³ μm²;

L—distance from borehole wall, m;

L_h—half-length of fracture, mm;

m—damage coefficient, dimensionless;

M_s—molar mass of solute, kg/mol;

p—pore pressure, MPa;

p₀—initial pore pressure of shale, MPa;

p_df—drilling fluid column pressure, MPa;

p_s—pore pressure of sealing layer, MPa;

r—distance from borehole center, m;

r_w—wellbore radius, m;

R—universal gas constant, 8.314 J/ (mol•K);

t—time, s;

T—absolute temperature, K;

x—distance from fracture center in x direction, m;

α—Biot coefficient, dimensionless;

α_d—correction coefficient of residual swelling pressure, dimensionless;

α_π—diffusion correction coefficient, dimensionless;

γ—damage coefficient, m³/mol;

Δy(x)—fracture width at x, m;

δ_i,j—coefficient with value 1 when i=j, and with value 0 when i≠j, dimensionless;

${{\varepsilon }_{\text{v0,chem}}}$—average hydration swelling volume strain, dimensionless;

${{\varepsilon }_{\text{v,chem}}}$—hydration swelling volume strain, dimensionless;

η—drilling fluid viscosity, mPa•s;

μ—chemical potential, J/mol;

μ_k—chemical potential of k component of pore fluid, J/mol;

ν—Poisson's ratio, dimensionless;

π_d—residual hydration swelling pressure in clay spacing, MPa;

${{\rho }_{\text{f}}}$—drilling fluid density, kg/m³;

${{\rho }_{\text{w}}}$—water density, kg/m³;

${{\sigma }_{\text{c}}}$—confining pressure, MPa;

${{\sigma }_{\text{e,}i\text{,}j}}$—component of effective stress, MPa;

${{\sigma }_{\text{h}}}$—minimum horizontal stress, MPa;

${{\sigma }_{\text{he}}}$—hydration stress, MPa;

${{\sigma }_{\text{hf}}}$—hydration swelling pressure in fracture, MPa;

${{\sigma }_{\text{H}}}$—maximum horizontal stress, MPa;

${{\sigma }_{i\text{,}j}}$—component of stress tensor, MPa;

${{\sigma }_{\text{m}}}$—shale membrane efficiency, dimensionless;

${{\sigma }_{\text{ms}}}$—sealing layer membrane efficiency, dimensionless;

${{\sigma }_{\text{t}}}$—tensile strength, MPa;

${{\sigma }_{\text{v}}}$—overburden pressure, MPa;

$\sigma _{1}^{*}$—maximum principal effective stress, MPa;

$\sigma _{3}^{*}$—minimum principal effective stress, MPa;

$\sigma _{i\text{,}j}^{*}$—effective stress considering strength damage, MPa;

τ—geometric characteristic parameter of pore, dimensionless;

ϕ—porosity, %;

${{\phi }_{\text{s}}}$—sealing layer porosity, %;

φ—internal friction angle of shale, (°);

ω—osmotic efficiency in clay spacing, dimensionless;

${{\omega }_{\text{0}}}$—hydration swelling coefficient related to solute concentration variation, MPa•s•m³/mol;

${{\omega }_{\text{1}}}$—damage coefficient related to solute concentration, m³/mol;

${{\omega }_{k}}$—hydration swelling coefficient related to chemical potential of k component, MPa•mol/J;

${{\omega }_{\text{k0}}}$—hydration swelling coefficient related to chemical potential of single component, MPa•mol/J.